Multipartite correlations and complementarity

Chiara Macchiavello University of Pavia

L. Maccone (University of Pavia) D. Bruss (University of Duesseldorf) B. Kraus, D. Sauerwein (University of Innsbruck)

OUTLINE

Provide interpretation of entanglement with classical correlations of measurement outcomes for complementary properties

Figures of merit: mutual information, Pearson coefficient

Role of quantum correlations and best separable states

Improvements: more than 2 complementary properties (qubits)

Generalisation to multipartite case

Entanglement

For two systems: pure states

mixed states

Previous approaches to entanglement based on non-locality, violation of Bell inequalities, positive partial transpose (PPT) criterion, etc

Here we focus on correlations of measurement outcomes for complementary observables

Reminder: complementary properties

If one knows the value of one property, all possible values of the other property are equally likely

E.g.: X, Y, Z for qubits

Simple examples

Maximally entangled state:

has perfect correlations both on 0/1 and +/-

Separable state

has perfect correlations for 0/1 and no correlation for +/-

Complementary correlations

Bipartite system

Compute classical correlations of measurement outcomes A-B and C-D

What figure of merit?

On system 1 measure either A or C

On system 2 measure either B or D

Mutual information

Complementary correlations

1)

2)

Complementary correlations for separable states

separable states entangled states

Do separable states achieve the bound? What kind of separable states, classically or quantum correlated?

Complementary correlations based on I are not convex and therefore they are not necessarily maximised on the border

Other separable states, with quantum correlations without entanglement, always lead to lower values

Optimal states are classically correlated (CC):

Complementary correlations for separable states

CQ and QC states:

QQ states:

Results

Two qubit states

Entanglement threshold

p=0,1 classically correlated otherwise q correlated

entangled for

entangled for

Pearson coefficient

Complementary correlations

perfect correlations

for product states

Conjecture 2)

1)

Results

Entanglement threshold

Notice: for separable states Person correlations are maximised by , classical and q correlations give the same value

Pearson vs mutual information

Even if the Pearson correlation measures only linear correlations, it is not weaker than information correlation:

The two-qubit state:

has negligible I for but Pearson correlations always >1!!

Improvement: 3 complementary properties

Add third complementary property: E and F (three Pauli operators for qubits)

Separable states fulfill

proved

conjectured

All entangled Werner states now exceed threshold of Pearson correlations

Multipartite correlation measure n: number of subsystems

N: number of complementary properties (MUBs)

Optimal states (maximal correlation measure)

1) Bipartite systems (n=2) with equal dimension d: maximally entangled states

2) Multipartite systems (n>2) with equal dimension: they can be mixed, e.g.

Info between one system and the rest

Pure multipartite states

2) Stabilised by local mutually unbiased operators

If a pure state has the following properties:

1) Completely mixed single-subsystems reduced states:

it has maximal correlation measure (sufficient condition)

Examples

1) n=3, d=2 QX

C2

Qz

Qy

3-tangle

2) n=3, d=3

3) n=d, d

Aharonov states

Correlation measure maximal in 2 bases, not in 3!

Correlation measure maximal in any N!

Correlation measure maximal in any N!

Entanglement detection

Starting from entropic uncertainty relations:

If the state is not fully separable

If the state is tripartite entangled

Tripartite entanglement detection

Eg: mixtures of GHZ and maximally mixed states:

Define:

Threshold value for p increases with increasing dimension

tripartite entanglement

SUMMARY AND OUTLOOK We introduced a classical info-theoretic approach to interpret entanglement by complementary observables

If correlations of measurement outcomes of complementary observables exceed a threshold value the state is entangled

Behaviour of classically and quantum correlated separable states depeds on the figure of merit (for I optimal separable states are CC) Improvement in the efficiency by adding a third complementary property (qubits)

Bipartite systems: maximal complementary correlations are necessary and sufficient condition for maximal entanglement

L. Maccone, D. Bruss & C.M., Phys. Rev. Lett. 114, 130401 (2015)

Multipartite systems: generalized correlation measures for any number of systems and complementary properties States that maximise complementary correlations are not necessarily pure D. Sauerwein, C.M., L. Maccone & B. Kraus, Phys. Rev. A 95, 042315 (2017)

Easy to implement! Z. Huang, L. Maccone, A. Karim, C.M.,R.J. Chapman & A. Peruzzo, Sci. Rep. 6, 27637 (2016)